This step is not the source of the fallacy.

However, you HAVE found a mistake!

There's a mistake in this step of the proof because the square root symbol
only has an unambiguous meaning when applied to positive numbers.

When x is positive, it has two square roots: one positive, and one
negative. By convention, the square root symbol
is defined to mean the positive one.

But that convention won't work when x is a negative number. For
instance, the two square roots of -1 are i and -i; these
cannot be distinguished on the basis of "positive" and "negative",
so how do we know which one is being meant by ?

Therefore, it's not clear what is being meant by this step of the proof.

However, this mistake is not the source of the fallacy. The mistake can
be corrected simply by specifying which square root is meant, for instance,
saying that

when x is negative, we are using the notation
to stand for the square root which is a positive
multiple of i, rather than the other one which is a negative multiple
of i.

Now it is unambiguously clear which square root is being referred to,
and that fixes up the mistake in this step of the proof.
One final comment: you may be tempted to think that the convention
of the square root symbol referring to the positive root still applies
here, thinking for example of i as the positive square root of -1
and -i as the negative one. This way of thinking is wrong,
because i is no more positive than -i is.

To understand this,
imagine a different culture where they use the symbol "j" for what we
call "-i", and they use "-j" for what we call "i". There is nothing
that makes our terminology any better than theirs; i and -i cannot
be distinguished by any arithmetical properties.

In their culture, they'd probably still use to mean the positive root when x is positive, just like
us (positive roots have different arithmetical properties than the
negative ones do, so their culture and ours would agree about which
is positive and which is negative).
But they might
adopt the opposite convention from us when x is negative, using
to mean j (our -i).

So the convention of letting refer to the positive root when x is positive does not
necessarily imply any convention about what it should refer to when x
is negative. That's why it's necessary to say in the proof which square
root is being meant.

Why don't you go back to the list of steps in the
proof and see if you can identify which one
is wrong, now that you know it isn't this one?
This page last updated: May 26, 1998Original Web Site Creator / Mathematical Content Developer:
Philip SpencerCurrent Network Coordinator and Contact Person:
Joel Chan - mathnet@math.toronto.edu